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<span><div><div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">我保证这篇文章和你以前看过的所有文章都不同，这是 2012 年还在果壳的时候写的，但是当时没有来得及写完就出国了……于是拖了两年，嗯，我是拖延症患者……</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">这篇文章的核心思想就是：</span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: bold; line-height: 1.5;">要让读者在不看任何数学公式的情况下理解傅里叶分析。</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">傅里叶分析不仅仅是一个数学工具，更是一种可以彻底颠覆一个人以前世界观的思维模式。但不幸的是，傅里叶分析的公式看起来太复杂了，所以很多大一新生上来就懵圈并从此对它深恶痛绝。老实说，这么有意思的东西居然成了大学里的杀手课程，不得不归咎于编教材的人实在是太严肃了。（您把教材写得好玩一点会死吗？会死吗？）所以我一直想写一个有意思的文章来解释傅里叶分析，有可能的话高中生都能看懂的那种。所以，不管读到这里的您从事何种工作，我保证您都能看懂，并且一定将体会到通过傅里叶分析看到世界另一个样子时的快感。至于对于已经有一定基础的朋友，也希望不要看到会的地方就急忙往后翻，仔细读一定会有新的发现。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">————以上是定场诗————</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">下面进入正题：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">抱歉，还是要啰嗦一句：其实学习本来就不是易事，我写这篇文章的初衷也是希望大家学习起来更加轻松，充满乐趣。但是千万！千万不要把这篇文章收藏起来，或是存下地址，心里想着：以后有时间再看。这样的例子太多了，也许几年后你都没有再打开这个页面。无论如何，耐下心，读下去。这篇文章要比读课本要轻松、开心得多……</span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: bold; line-height: 1.5;">一、什么是频域</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">从我们出生，我们看到的世界都以时间贯穿，股票的走势、人的身高、汽车的轨迹都会随着时间发生改变。这种以时间作为参照来观察动态世界的方法我们称其为时域分析。而我们也想当然的认为，世间万物都在随着时间不停的改变，并且永远不会静止下来。但如果我告诉你，用另一种方法来观察世界的话，你会发现</span><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: bold;">世界是永恒不变的</span><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">，你会不会觉得我疯了？我没有疯，这个静止的世界就叫做频域。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal; font-weight: bold;">先举一个公式上并非很恰当，但意义上再贴切不过的例子：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">在你的理解中，一段音乐是什么呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="font-size: 9pt;"><img src="傅里叶变换_files/7cc829d3gw1eh5v4r7dlkj20go05mwff.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v4r7dlkj20go05mwff.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">这是我们对音乐最普遍的理解，一个随着时间变化的震动。但我相信对于乐器小能手们来说，音乐更直观的理解是这样的：</span></div><p style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"></p><div><img src="傅里叶变换_files/7cc829d3gw1eh5v4w5m0xj20go05nwff.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v4w5m0xj20go05nwff.jpg"/></div><div style="margin-top: 1em; margin-bottom: 1em;"><span style="-en-paragraph:true;">好的！下课，同学们再见。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">是的，其实这一段写到这里已经可以结束了。上图是音乐在时域的样子，而下图则是音乐在频域的样子。所以频域这一概念对大家都从不陌生，只是从来没意识到而已。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">现在我们可以回过头来重新看看一开始那句痴人说梦般的话：世界是永恒的。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; font-size: 12px; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-caps: normal; font-variant-ligatures: normal;">将以上两图简化：</span></div></div><div><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">时域：</span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v4xmosuj20go096t9l.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v4xmosuj20go096t9l.jpg"/></span></div><div><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">频域：</span></div><div><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1egu4n43a2gj203t05jq2r.jpg" type="image/jpeg" data-filename="7cc829d3gw1egu4n43a2gj203t05jq2r.jpg"/></span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在时域，我们观察到钢琴的琴弦一会上一会下的摆动，就如同一支股票的走势；而在频域，只有那一个永恒的音符。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">所以</span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">你眼中看似落叶纷飞变化无常的世界，实际只是躺在上帝怀中一份早已谱好的乐章。</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">抱歉，这不是一句鸡汤文，而是黑板上确凿的公式：傅里叶同学告诉我们，任何周期函数，都可以看作是不同振幅，不同相位正弦波的叠加。在第一个例子里我们可以理解为，利用对不同琴键不同力度，不同时间点的敲击，可以组合出任何一首乐曲。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">而贯穿时域与频域的方法之一，就是传中说的傅里叶分析。傅里叶分析可分为傅里叶级数（Fourier Serie）和傅里叶变换(Fourier Transformation)，我们从简单的开始谈起。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"> </span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">二、傅里叶级数(Fourier Series)的频谱</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">还是举个栗子并且有图有真相才好理解。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">如果我说我能用前面说的正弦曲线波叠加出一个带 90 度角的矩形波来，你会相信吗？你不会，就像当年的我一样。但是看看下图：</span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v4yfzcsj20go0cujt7.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v4yfzcsj20go0cujt7.jpg"/></span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">第一幅图是一个郁闷的正弦波 cos（x）</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">第二幅图是 2 个卖萌的正弦波的叠加 cos (x) +a.cos (3x)</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">第三幅图是 4 个发春的正弦波的叠加</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">第四幅图是 10 个便秘的正弦波的叠加</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">随着正弦波数量逐渐的增长，他们最终会叠加成一个标准的矩形，大家从中体会到了什么道理？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">（只要努力，弯的都能掰直！）</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">随着叠加的递增，所有正弦波中上升的部分逐渐让原本缓慢增加的曲线不断变陡，而所有正弦波中下降的部分又抵消了上升到最高处时继续上升的部分使其变为水平线。一个矩形就这么叠加而成了。但是要多少个正弦波叠加起来才能形成一个标准 90 度角的矩形波呢？不幸的告诉大家，答案是无穷多个。（上帝：我能让你们猜着我？）</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">不仅仅是矩形，你能想到的任何波形都是可以如此方法用正弦波叠加起来的。这是没有接触过傅里叶分析的人在直觉上的第一个难点，但是一旦接受了这样的设定，游戏就开始有意思起来了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">还是上图的正弦波累加成矩形波，我们换一个角度来看看：</span></span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v4zmdijj20go0qun0b.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v4zmdijj20go0qun0b.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在这几幅图中，最前面黑色的线就是所有正弦波叠加而成的总和，也就是越来越接近矩形波的那个图形。而后面依不同颜色排列而成的正弦波就是组合为矩形波的各个分量。这些正弦波按照频率从低到高从前向后排列开来，而每一个波的振幅都是不同的。一定有细心的读者发现了，每两个正弦波之间都还有一条直线，那并不是分割线，而是振幅为 0 的正弦波！也就是说，为了组成特殊的曲线，有些正弦波成分是不需要的。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这里，不同频率的正弦波我们成为频率分量。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); font-weight: bold;">好了，关键的地方来了！！</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">如果我们把第一个频率最低的频率分量看作“1”，我们就有了构建频域的最基本单元。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">对于我们最常见的有理数轴，数字“1”就是有理数轴的基本单元。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">（好吧，数学称法为——基。在那个年代，这个字还没有其他奇怪的解释，后面还有正交基这样的词汇我会说吗?）</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">时域的基本单元就是“1 秒”，如果我们将一个角频率为</span><img src="https://images0.cnblogs.com/news/66372/201405/271732411343005.png" style="margin: 0px; padding: 0px; border: 0px; max-width: 660px; height: auto;"></img><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">的正弦波 cos（</span><img src="https://images0.cnblogs.com/news/66372/201405/271732412442504.png" style="margin: 0px; padding: 0px; border: 0px; max-width: 660px; height: auto;"></img><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">t）看作基础，那么频域的基本单元就是</span><img src="https://images0.cnblogs.com/news/66372/201405/271732416978431.png" style="margin: 0px; padding: 0px; border: 0px; max-width: 660px; height: auto;"></img><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">有了“1”，还要有“0”才能构成世界，那么频域的“0”是什么呢？cos（0t）就是一个周期无限长的正弦波，也就是一条直线！所以在频域，0 频率也被称为直流分量，在傅里叶级数的叠加中，它仅仅影响全部波形相对于数轴整体向上或是向下而不改变波的形状。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">接下来，让我们回到初中，回忆一下已经死去的八戒，啊不，已经死去的老师是怎么定义正弦波的吧。</span></span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1egu4n8nmvhg20fk05lmxo.gif" type="image/gif" data-filename="7cc829d3gw1egu4n8nmvhg20fk05lmxo.gif"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">正弦波就是一个圆周运动在一条直线上的投影。所以频域的基本单元也可以理解为一个始终在旋转的圆</span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1egu4mtx2sjg2074074tha.gif" type="image/gif" data-filename="7cc829d3gw1egu4mtx2sjg2074074tha.gif"/></span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/271734071031740.gif" type="image/gif" data-filename="271734071031740.gif"/></span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">介绍完了频域的基本组成单元，我们就可以看一看一个矩形波，在频域里的另一个模样了：</span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v5098jzj20go0btq3q.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v5098jzj20go0btq3q.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">这就是矩形波在频域的样子，是不是完全认不出来了？教科书一般就给到这里然后留给了读者无穷的遐想，以及无穷的吐槽，其实教科书只要补一张图就足够了：频域图像，也就是俗称的频谱，就是——</span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v53i4kjj20go07iab8.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v53i4kjj20go07iab8.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">再清楚一点：</span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v57q4vij20go0a0gn0.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v57q4vij20go0a0gn0.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">可以发现，在频谱中，偶数项的振幅都是0，也就对应了图中的彩色直线。振幅为 0 的正弦波。</span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/271734446194512.gif" type="image/gif" data-filename="271734446194512.gif"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">老实说，在我学傅里叶变换时，维基的这个图还没有出现，那时我就想到了这种表达方法，而且，后面还会加入维基没有表示出来的另一个谱——相位谱。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">但是在讲相位谱之前，我们先回顾一下刚刚的这个例子究竟意味着什么。记得前面说过的那句“世界是静止的”吗？估计好多人对这句话都已经吐槽半天了。想象一下，世界上每一个看似混乱的表象，实际都是一条时间轴上不规则的曲线，但实际这些曲线都是由这些无穷无尽的正弦波组成。我们看似不规律的事情反而是规律的正弦波在时域上的投影，而正弦波又是一个旋转的圆在直线上的投影。那么你的脑海中会产生一个什么画面呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">我们眼中的世界就像皮影戏的大幕布，幕布的后面有无数的齿轮，大齿轮带动小齿轮，小齿轮再带动更小的。在最外面的小齿轮上有一个小人——那就是我们自己。我们只看到这个小人毫无规律的在幕布前表演，却无法预测他下一步会去哪。而幕布后面的齿轮却永远一直那样不停的旋转，永不停歇。这样说来有些宿命论的感觉。说实话，这种对人生的描绘是我一个朋友在我们都是高中生的时候感叹的，当时想想似懂非懂，直到有一天我学到了傅里叶级数……</span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">三、傅里叶级数（Fourier Series）的相位谱</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">上一章的关键词是：从侧面看。这一章的关键词是：从下面看。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在这一章最开始，我想先回答很多人的一个问题：傅里叶分析究竟是干什么用的？这段相对比较枯燥，已经知道了的同学可以直接跳到下一个分割线。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">先说一个最直接的用途。无论听广播还是看电视，我们一定对一个词不陌生——频道。频道频道，就是频率的通道，不同的频道就是将不同的频率作为一个通道来进行信息传输。下面大家尝试一件事：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">先在纸上画一个sin（x），不一定标准，意思差不多就行。不是很难吧。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">好，接下去画一个sin（3x）+sin（5x）的图形。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">别说标准不标准了，曲线什么时候上升什么时候下降你都不一定画的对吧？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">好，画不出来不要紧，我把sin（3x）+sin（5x）的曲线给你，但是前提是你不知道这个曲线的方程式，现在需要你把sin（5x）给我从图里拿出去，看看剩下的是什么。这基本是不可能做到的。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">但是在频域呢？则简单的很，无非就是几条竖线而已。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">所以很多在时域看似不可能做到的数学操作，在频域相反很容易。这就是需要傅里叶变换的地方。尤其是从某条曲线中去除一些特定的频率成分，这在工程上称为滤波，是信号处理最重要的概念之一，只有在频域才能轻松的做到。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">再说一个更重要，但是稍微复杂一点的用途——求解微分方程。（这段有点难度，看不懂的可以直接跳过这段）微分方程的重要性不用我过多介绍了。各行各业都用的到。但是求解微分方程却是一件相当麻烦的事情。因为除了要计算加减乘除，还要计算微分积分。而傅里叶变换则可以让微分和积分在频域中变为乘法和除法，大学数学瞬间变小学算术有没有。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">傅里叶分析当然还有其他更重要的用途，我们随着讲随着提。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">————————————————————————————————————</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">下面我们继续说相位谱：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">通过时域到频域的变换，我们得到了一个从侧面看的频谱，但是这个频谱并没有包含时域中全部的信息。因为频谱只代表每一个对应的正弦波的振幅是多少，而没有提到相位。基础的正弦波A.sin(wt+θ)中，振幅，频率，相位缺一不可，不同相位决定了波的位置，所以对于频域分析，仅仅有频谱（振幅谱）是不够的，我们还需要一个相位谱。那么这个相位谱在哪呢？我们看下图，这次为了避免图片太混论，我们用7个波叠加的图。</span></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v58ov24j20go0hutao.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v58ov24j20go0hutao.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">鉴于正弦波是周期的，我们需要设定一个用来标记正弦波位置的东西。在图中就是那些小红点。小红点是距离频率轴最近的波峰，而这个波峰所处的位置离频率轴有多远呢？为了看的更清楚，我们将红色的点投影到下平面，投影点我们用粉色点来表示。当然，这些粉色的点只标注了波峰距离频率轴的距离，并不是相位。</span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v5cn9yrj20go0fsgmu.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v5cn9yrj20go0fsgmu.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这里需要纠正一个概念：时间差并不是相位差。如果将全部周期看作2Pi或者360度的话，相位差则是时间差在一个周期中所占的比例。我们将时间差除周期再乘2Pi，就得到了相位差。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在完整的立体图中，我们将投影得到的时间差依次除以所在频率的周期，就得到了最下面的相位谱。所以，频谱是从侧面看，相位谱是从下面看。下次偷看女生裙底被发现的话，可以告诉她：“对不起，我只是想看看你的相位谱。”</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">注意到，相位谱中的相位除了0，就是Pi。因为cos（t+Pi）=-cos（t），所以实际上相位为Pi的波只是上下翻转了而已。对于周期方波的傅里叶级数，这样的相位谱已经是很简单的了。另外值得注意的是，由于cos（t+2Pi）=cos（t），所以相位差是周期的，pi和3pi，5pi，7pi都是相同的相位。人为定义相位谱的值域为(-pi，pi]，所以图中的相位差均为Pi。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">最后来一张大集合：</span></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5v5dhli2j20go0dtwga.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5v5dhli2j20go0dtwga.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">四、傅里叶变换（Fourier Tranformation）</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">相信通过前面三章，大家对频域以及傅里叶级数都有了一个全新的认识。但是文章在一开始关于钢琴琴谱的例子我曾说过，这个栗子是一个公式错误，但是概念典型的例子。所谓的公式错误在哪里呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">傅里叶级数的本质是将一个周期的信号分解成无限多分开的（离散的）正弦波，但是宇宙似乎并不是周期的。曾经在学数字信号处理的时候写过一首打油诗：</span></div><h3 style="margin: 10px 0px; padding: 0px; font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">往昔连续非周期，</span></h3><h3 style="margin: 10px 0px; padding: 0px; font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">回忆周期不连续，</span></h3><h3 style="margin: 10px 0px; padding: 0px; font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">任你ZT、DFT，</span></h3><h3 style="margin: 10px 0px; padding: 0px; font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 16px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">还原不回去。</span></h3><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">（请无视我渣一样的文学水平……）</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在这个世界上，有的事情一期一会，永不再来，并且时间始终不曾停息地将那些刻骨铭心的往昔连续的标记在时间点上。但是这些事情往往又成为了我们格外宝贵的回忆，在我们大脑里隔一段时间就会周期性的蹦出来一下，可惜这些回忆都是零散的片段，往往只有最幸福的回忆，而平淡的回忆则逐渐被我们忘却。因为，往昔是一个连续的非周期信号，而回忆是一个周期离散信号。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">是否有一种数学工具将连续非周期信号变换为周期离散信号呢？抱歉，真没有。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">比如傅里叶级数，在时域是一个周期且连续的函数，而在频域是一个非周期离散的函数。这句话比较绕嘴，实在看着费事可以干脆回忆第一章的图片。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">而在我们接下去要讲的傅里叶变换，则是将一个时域非周期的连续信号，转换为一个在频域非周期的连续信号。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">算了，还是上一张图方便大家理解吧：</span></span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2g0nhuj20go08baan.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2g0nhuj20go08baan.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">或者我们也可以换一个角度理解：傅里叶变换实际上是对一个周期无限大的函数进行傅里叶变换。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">所以说，钢琴谱其实并非一个连续的频谱，而是很多在时间上离散的频率，但是这样的一个贴切的比喻真的是很难找出第二个来了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">因此在傅里叶变换在频域上就从离散谱变成了连续谱。那么连续谱是什么样子呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); font-weight: bold;">你见过大海么？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">为了方便大家对比，我们这次从另一个角度来看频谱，还是傅里叶级数中用到最多的那幅图，我们从频率较高的方向看。</span></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2gl2u3j20go07z756.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2gl2u3j20go07z756.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">以上是离散谱，那么连续谱是什么样子呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">尽情的发挥你的想象，想象这些离散的正弦波离得越来越近，逐渐变得连续……</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">直到变得像波涛起伏的大海：</span></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2i13r8j20go08smyb.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2i13r8j20go08smyb.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">很抱歉，为了能让这些波浪更清晰的看到，我没有选用正确的计算参数，而是选择了一些让图片更美观的参数，不然这图看起来就像屎一样了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">不过通过这样两幅图去比较，大家应该可以理解如何从离散谱变成了连续谱的了吧？原来离散谱的叠加，变成了连续谱的累积。所以在计算上也从求和符号变成了积分符号。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">不过，这个故事还没有讲完，接下去，我保证让你看到一幅比上图更美丽壮观的图片，但是这里需要介绍到一个数学工具才能然故事继续，这个工具就是——</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"> </span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">五、宇宙耍帅第一公式：欧拉公式</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">虚数i这个概念大家在高中就接触过，但那时我们只知道它是-1 的平方根，可是它真正的意义是什么呢?</span></span><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2kqrs0j20go09iaaq.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2kqrs0j20go09iaaq.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">同时，我们获得了一个垂直的虚数轴。实数轴与虚数轴共同构成了一个复数的平面，也称复平面。这样我们就了解到，乘虚数i的一个功能——旋转。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">现在，就有请宇宙第一耍帅公式欧拉公式隆重登场——</span></div><p style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"></p><div style="text-align: left;"><img src="傅里叶变换_files/071037530204619.png" type="image/png" data-filename="071037530204619.png"/><br/></div><div style="margin-top: 1em; margin-bottom: 1em; text-align: left;"><span style="-en-paragraph:true;">这个公式在数学领域的意义要远大于傅里叶分析，但是乘它为宇宙第一耍帅公式是因为它的特殊形式——当x等于 Pi 的时候。</span></div><p style="text-align: left;"></p><p style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"></p><div style="text-align: left;"><img src="傅里叶变换_files/071037530679604.png" type="image/png" data-filename="071037530679604.png"/><br/></div><div style="margin-top: 1em; margin-bottom: 1em; text-align: left;"><span style="-en-paragraph:true;">经常有理工科的学生为了跟妹子表现自己的学术功底，用这个公式来给妹子解释数学之美：”石榴姐你看，这个公式里既有自然底数e，自然数 1 和0，虚数i还有圆周率 pi，它是这么简洁，这么美丽啊！“但是姑娘们心里往往只有一句话：”臭屌丝……“</span></div><p style="text-align: left;"></p><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="background-color:rgb(255, 255, 255);color:rgb(0, 0, 0);font-family:Verdana, Arial, Helvetica, sans-serif;font-size:12px;font-variant-caps:normal;font-variant-ligatures:normal;"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">这个公式关键的作用，是将正弦波统一成了简单的指数形式。我们来看看图像上的涵义：</span></span><img src="傅里叶变换_files/7cc829d3gw1eh5q2kwv90j20go08b0tx.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2kwv90j20go08b0tx.jpg"/><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">欧拉公式所描绘的，是一个随着时间变化，在复平面上做圆周运动的点，随着时间的改变，在时间轴上就成了一条螺旋线。如果只看它的实数部分，也就是螺旋线在左侧的投影，就是一个最基础的余弦函数。而右侧的投影则是一个正弦函数。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">关于复数更深的理解，大家可以参考：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><a href="http://www.zhihu.com/question/23234701/answer/26017000" style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); text-decoration: underline;">复数的物理意义是什么？</a></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这里不需要讲的太复杂，足够让大家理解后面的内容就可以了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"> </span></div><h2 style="margin: 10px 0px; padding: 0px; font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="font-size: 21px; font-weight: bold; line-height: 1.5; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">六、指数形式的傅里叶变换</span></h2><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">有了欧拉公式的帮助，我们便知道：正弦波的叠加，也可以理解为螺旋线的叠加在实数空间的投影。而螺旋线的叠加如果用一个形象的栗子来理解是什么呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255); font-weight: bold;">　　光波</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">高中时我们就学过，自然光是由不同颜色的光叠加而成的，而最著名的实验就是牛顿师傅的三棱镜实验：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2na8puj207p04uwef.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2na8puj207p04uwef.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">所以其实我们在很早就接触到了光的频谱，只是并没有了解频谱更重要的意义。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">但不同的是，傅里叶变换出来的频谱不仅仅是可见光这样频率范围有限的叠加，而是频率从 0 到无穷所有频率的组合。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这里，我们可以用两种方法来理解正弦波：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">第一种前面已经讲过了，就是螺旋线在实轴的投影。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">另一种需要借助欧拉公式的另一种形式去理解：</span></div><p style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"></p><div><img src="傅里叶变换_files/071037533339291.png" type="image/png" data-filename="071037533339291.png"/></div><div style="margin-top: 1em; margin-bottom: 1em;"><img src="傅里叶变换_files/071037541452176.png" type="image/png" data-filename="071037541452176.png"/><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">将以上两式相加再除2，得到：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2uzxlfj203k010t8h.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2uzxlfj203k010t8h.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这个式子可以怎么理解呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">我们刚才讲过，e^(it)可以理解为一条逆时针旋转的螺旋线，那么e^(-it)则可以理解为一条顺时针旋转的螺旋线。而 cos (t)则是这两条旋转方向不同的螺旋线叠加的一半，因为这两条螺旋线的虚数部分相互抵消掉了！</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">举个例子的话，就是极化方向不同的两束光波，磁场抵消，电场加倍。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这里，逆时针旋转的我们称为正频率，而顺时针旋转的我们称为负频率（注意不是复频率）。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">好了，刚才我们已经看到了大海——连续的傅里叶变换频谱，现在想一想，连续的螺旋线会是什么样子：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">想象一下再往下翻：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2psb9wj20m80h8wi6.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2psb9wj20m80h8wi6.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">是不是很漂亮？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">你猜猜，这个图形在时域是什么样子？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2sq0eqj20go0d33z1.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2sq0eqj20go0d33z1.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">哈哈，是不是觉得被狠狠扇了一个耳光。数学就是这么一个把简单的问题搞得很复杂的东西。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">顺便说一句，那个像大海螺一样的图，为了方便观看，我仅仅展示了其中正频率的部分，负频率的部分没有显示出来。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">如果你认真去看，海螺图上的每一条螺旋线都是可以清楚的看到的，每一条螺旋线都有着不同的振幅（旋转半径），频率（旋转周期）以及相位。而将所有螺旋线连成平面，就是这幅海螺图了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">好了，讲到这里，相信大家对傅里叶变换以及傅里叶级数都有了一个形象的理解了，我们最后用一张图来总结一下：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;"><span style="font-size: 9pt; font-family: Verdana;"><img src="傅里叶变换_files/7cc829d3gw1eh5q2t82oxj20go0r8acj.jpg" type="image/jpeg" data-filename="7cc829d3gw1eh5q2t82oxj20go0r8acj.jpg"/></span><br/></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">好了，傅里叶的故事终于讲完了，下面来讲讲我的故事：</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这篇文章第一次被卸下来的地方你们绝对猜不到在哪，是在一张高数考试的卷子上。当时为了刷分，我重修了高数（上），但是后来时间紧压根没复习，所以我就抱着裸考的心态去了考场。但是到了考场我突然意识到，无论如何我都不会比上次考的更好了，所以干脆写一些自己对于数学的想法吧。于是用了一个小时左右的时间在试卷上洋洋洒洒写了本文的第一草稿。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">你们猜我的了多少分？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">6 分</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">没错，就是这个数字。而这 6 分的成绩是因为最后我实在无聊，把选择题全部填上了C，应该是中了两道，得到了这宝贵的 6 分。说真的，我很希望那张卷子还在，但是应该不太可能了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">那么你们猜猜我第一次信号与系统考了多少分呢？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">45 分</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">没错，刚刚够参加补考的。但是我心一横没去考，决定重修。因为那个学期在忙其他事情，学习真的就抛在脑后了。但是我知道这是一门很重要的课，无论如何我要吃透它。说真的，信号与系统这门课几乎是大部分工科课程的基础，尤其是通信专业。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在重修的过程中，我仔细分析了每一个公式，试图给这个公式以一个直观的理解。虽然我知道对于研究数学的人来说，这样的学习方法完全没有前途可言，因为随着概念愈加抽象，维度越来越高，这种图像或者模型理解法将完全丧失作用。但是对于一个工科生来说，足够了。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">后来来了德国，这边学校要求我重修信号与系统时，我彻底无语了。但是没办法，德国人有时对中国人就是有种藐视，觉得你的教育不靠谱。所以没办法，再来一遍吧。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这次，我考了满分，而及格率只有一半。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">老实说，数学工具对于工科生和对于理科生来说，意义是完全不同的。工科生只要理解了，会用，会查，就足够了。但是很多高校却将这些重要的数学课程教给数学系的老师去教。这样就出现一个问题，数学老师讲得天花乱坠，又是推理又是证明，但是学生心里就只有一句话：学这货到底干嘛用的？</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">缺少了目标的教育是彻底的失败。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">在开始学习一门数学工具的时候，学生完全不知道这个工具的作用，现实涵义。而教材上有只有晦涩难懂，定语就二十几个字的概念以及看了就眼晕的公式。能学出兴趣来就怪了！</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">好在我很幸运，遇到了大连海事大学的吴楠老师。他的课全程来看是两条线索，一条从上而下，一条从下而上。先将本门课程的意义，然后指出这门课程中会遇到哪样的问题，让学生知道自己学习的某种知识在现实中扮演的角色。然后再从基础讲起，梳理知识树，直到延伸到另一条线索中提出的问题，完美的衔接在一起！</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">这样的教学模式，我想才是大学里应该出现的。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">最后，写给所有给我点赞并留言的同学。真的谢谢大家的支持，也很抱歉不能一一回复。因为知乎专栏的留言要逐次加载，为了看到最后一条要点很多次加载。当然我都坚持看完了，只是没办法一一回复。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);">本文只是介绍了一种对傅里叶分析新颖的理解方法，对于求学，还是要踏踏实实弄清楚公式和概念，学习，真的没有捷径。但至少通过本文，我希望可以让这条漫长的路变得有意思一些。</span></div><div style="margin: 10px auto; padding: 0px; text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-align: left; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; background-color: rgb(255, 255, 255);"><span style="text-indent: 0px; color: rgb(0, 0, 0); font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 12px; font-variant-ligatures: normal; font-variant-caps: normal; letter-spacing: normal; orphans: 2; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;">最后，祝大家都能在学习中找到乐趣…</span></div></div></span>
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